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Kalman decomposition

Mathematical Theory From Wikipedia, the free encyclopedia

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In control theory, a Kalman decomposition provides a mathematical means to convert a representation of any linear time-invariant (LTI) control system to a form in which the system can be decomposed into a standard form which makes clear the observable and controllable components of the system. This decomposition results in the system being presented with a more illuminating structure, making it easier to draw conclusions on the system's reachable and observable subspaces.

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Definition

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Consider the continuous-time LTI control system

,
,

or the discrete-time LTI control system

,
.

The Kalman decomposition is defined as the realization of this system obtained by transforming the original matrices as follows:

,
,
,
,

where is the coordinate transformation matrix defined as

,

and whose submatrices are

  •  : a matrix whose columns span the subspace of states which are both reachable and unobservable.
  •  : chosen so that the columns of are a basis for the reachable subspace.
  •  : chosen so that the columns of are a basis for the unobservable subspace.
  •  : chosen so that is invertible.

It can be observed that some of these matrices may have dimension zero. For example, if the system is both observable and controllable, then , making the other matrices zero dimension.

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Consequences

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By using results from controllability and observability, it can be shown that the transformed system has matrices in the following form:

This leads to the conclusion that

  • The subsystem is both reachable and observable.
  • The subsystem is reachable.
  • The subsystem is observable.
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Variants

A Kalman decomposition also exists for linear dynamical quantum systems. Unlike classical dynamical systems, the coordinate transformation used in this variant requires to be in a specific class of transformations due to the physical laws of quantum mechanics.[1]

See also

References

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